We examine the formation and growth of isolated myelin figures and microscale multilamellar tubules from isotropic micellar solutions of an anionic surfactant. Upon cooling, surfactant micelles transform into multilamellar vesicles (MLVs) whose contact is found to trigger the unidirectional growth of myelins. While the MLV diameter grows as dMLV ∝ t1/2, myelins grow linearly in time as LM ∝ t1, with a fixed diameter. Combining time-resolved small-angle neutron scattering (SANS) and optical microscopy, we demonstrate that the microscopic growth of spherical MLVs and cylindrical myelins stems from the same nanoscale molecular mechanism, namely, the surfactant exchange from micelles into curved lamellar structures at a constant volumetric rate. This mechanism successfully describes the growth rate of (nonequilibrium) myelin figures based on a population balance at thermodynamic equilibrium.